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Math SATs- Sequences and Series

Math SATs- Sequences and Series

Welcome to a Math SAT Preparation Lesson. In this lesson we will study sequences and series.

Let’s start off with sequences

A sequence is a list of numbers that has a certain order. For example, 16,7,888,-2,0,72 is a sequence, since 16 is the first number in the list, 7 is the second, etc. Sequences can have a certain “rule” by which terms progress, but they can also be completely random.

A few examples of cool sequences I can think of off the top of my head:

1,4,9,16… a sequence of the perfect squares starting from 1.

1,1,2,3,5,8,13,21,34,... the Fibonacci sequence, where each term is the sum of the previous two terms.

1, \frac{1}{2}, \frac{1}{3},... the harmonic sequence.

0,0,0,0… the “empty” sequence.

There are 2 special types of sequences we like to look at- arithmetic ones and geometric ones.

Arithmetic sequences are sequences that start with any number a, and in which every n^ {th} term can be written as a_n=a+(n-1)d, where d is any number. An example of such a sequence would be 5, 12, 19, 26, 33…, where a=5 and d=7. This is an increasing arithmetic sequence, as the terms are increasing. Decreasing arithmetic sequences have d<0.

Geometric sequences also start with any number a (though usually a is nonzero here), but this time we are not adding an extra d value each time- we multiply a by a factor of r. Thus, the n ^ {th} term is a_n=ar ^ {n-1}. Geometric sequences can either be monotonic, when r is positive and the terms are moving in one direction, or alternating, where r<0 and the terms alternate between positive and negative values, depending on n.

Sequences are fun, but they have an important application- series.

A series is a sequence of numbers that represent partial sums for another sequence. For example, if my sequence is 1,2,3,4… then my series would be 1,1+2,1+2+3,..., or 1,3,6,10….

With arithmetic and geometric series, we can use a formula to calculate any n^{th} term of the series. These shortcuts are useful since they save you from having to write out the entire sequence and add all of its terms up.

The formula for an arithmetic series is:

s_n=\frac{n(2a+(n-1)d)}{2}

The formula for a geometric series is:

s_n\frac{a(1-r ^ {n+1})}{1-r}

These formulas are worth memorizing.

The trick in solving sequence and series problems is recognizing first the type of sequence you’re dealing with, and then finding the proper a, d or r if you’re dealing with an arithmetic or geometric ones.

Are you a king of sequences?

Can you spot the patterns and calculate the n ^ {th} term? Check out the Sequence and Series test.

arjun7391
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arjun7391 said:

ya its kool>> i like it…. but wheres da harmonic mean in detail>> n only frm these formulas will b asked or what….....

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  • Posted 2 months ago.
oLahav
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oLahav said in response to:
arjun7391
arjun7391’s post:
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ya its kool>> i like it…. but wheres da harmonic mean in detail>> n only frm these formulas will b asked or what….....

The harmonic sequence just means that for each n from 1 to infinity, the nth term of the sequence is 1/n.

On the SATs you may be asked about any sort of sequence- arithmatic, geometric, even some random ones you’ll have to spot yourself. The formulas given here are most frequently asked about and they’ll also help you in your later studies of sequences and series in terms of calculus limits, if you go on to study math at college, so I thought including them is a good idea.

For any other questions, post a discussion. And make sure to check out the Sequences test to know exactly what type of questions you’ll be asked.

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  • Posted 2 months ago.
sami
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sami said:

haronic mean is not in detail

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  • Posted 26 days ago.
Geetika
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Geetika said:

Hi Guys, I m searching for Daily updated questions on diferent subjects. If anyine knows where I can get those Questions, Let me know. Thanks

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  • Posted 23 days ago.
oLahav
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oLahav said in response to:
Geetika
Geetika’s post:
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Hi Guys, I m searching for Daily updated questions on diferent subjects. If anyine knows where I can get those Questions, Let me know. Thanks

You can find a lot of questions in our Question Bank, which has hundreds of original SAT questions on everything from geometry to sentence corrections.

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  • Posted 23 days ago.
kausar
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kausar said:

thats a good and easy to learn part of serires, but it is beter to give some detail and multiple examples

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  • Posted 15 days ago.
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